To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as having two stable states: in alignment with a magnetic field, or in anti-alignment. and because the magnetic field is a measure of some kind of rotation it can add to or subtract from the angular momentum of a spinor. this difference is "felt" as a negative or positive potential difference. this you can think of as two opposite forces on the spinor that split it apart into up and down components. the interesting thing is that a spinor with an arbitrary axis can always be written as the sum/superposition of an up and down spinor for some chosen direction. turns out quaternions have precisely the properties that you need to model this.
i hope this was intelligible, it's a bit hard to put the geometry into words.
Woah. Ok, that model totally blew my mind. I never understood what angular momentum had anything to do with it, but the gyroscope analogy makes total sense. Thank you!
Unfortunately i can't. I've been thinking about spinors and related concepts as a hobby for maybe two years now. But i never found an explanation like mine above anywhere. Good geometric intuition about these things does not seem to be highly valued in physics education so people don't seek it. It turns out you can represent a dirac spinor (as the sum of left and right weyl spinors) extremely well just with your two hands. the dirac matrices simply tell you how to rotate your hands or mirror them (which you can do because you have two!). I should make a video about this actually...
